They're making it very clear that they gave no intention of giving you right of way, so I doubt you'll get a polite response. Where is the gate? Is it against the bottom of the garden or do you have to walk close to their house?

YWNBU, you have right of access and they are blocking that access. Give them a gentle reminder and say you'd hate to damage their bike/gate by crashing into it with your bins when you are putting them out

Have a word and just politely explain that you have to cross their yard to empty your bin. Explain that you have no desire to invade their privacy but you do have the legal right and necessity to do this so how can you make it least painful for them.